Meshfree, or particle methods, offer many numerical advantages over conventional finite element and finite difference methods in modeling large deformation and moving discontinuity problems in solid and structural applications. Those methods were also found to be very effective in reducing the volumetric locking and shear locking in the solid and structural analyses. The earliest development in meshfree methods was the Smoothed Particle Hydrodynamics (SPH) method or one of the Galerkin-based meshfree methods. In this method, partial differential equations are transformed into integral equations and the kernel estimate then provides the approximation to estimate the field variables at discrete particles. Since the functions are evaluated only at particles, the use of a mesh is no longer required. The ability to handle severe deformations without the use of meshes in fluid-like motion allows SPH method to be applied to problems that historically have been reserved for Eulerian approaches.
Numerical simulation of a structure made of brittle material subjected to cracks is one of the computational challenges when using Galerkin-based meshfree methods. Local and non-local strain fields are simultaneously presented in a meshfree model as a result of cracks in the structure. Other challenges such as the presence of spurious or zero-energy modes in SPH or other Galerkin-based meshfree methods also exist due to the rank instability caused by the under-integration of the weak forms inherent in the central difference formula from nodal integration approach. Prior art approaches have been ad hoc and sometimes ineffective.
Therefore, it would be desirable to have improved meshfree methods of obtaining numerically simulated structural behaviors of brittle material based on damage mechanics.